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<div class="titlepage"><div><div><h4 class="title">
<a name="geometry.reference.concepts.concept_polygon"></a><a class="link" href="concept_polygon.html" title="Polygon Concept">Polygon
        Concept</a>
</h4></div></div></div>
<h6>
<a name="geometry.reference.concepts.concept_polygon.h0"></a>
          <span class="phrase"><a name="geometry.reference.concepts.concept_polygon.description"></a></span><a class="link" href="concept_polygon.html#geometry.reference.concepts.concept_polygon.description">Description</a>
        </h6>
<p>
          The Polygon Concept describes the requirements for a polygon type. All
          algorithms in Boost.Geometry will check any geometry arguments against
          the concept requirements.
        </p>
<p>
          A polygon is <span class="emphasis"><em>A polygon is a planar surface defined by one exterior
          boundary and zero or more interior boundaries</em></span> (<a href="http://www.opengeospatial.org/standards/sfa" target="_top">OGC
          Simple Feature Specification</a>).
        </p>
<p>
          So the definition of a Boost.Geometry polygon differs a bit from e.g. Wiki,
          where a polygon does not have holes. A polygon of Boost.Geometry is a polygon
          with or without holes. <span class="emphasis"><em>(A polygon without holes is a <span class="emphasis"><em>helper
          geometry</em></span> within Boost.Geometry, and referred to as a ring.)</em></span>
        </p>
<h6>
<a name="geometry.reference.concepts.concept_polygon.h1"></a>
          <span class="phrase"><a name="geometry.reference.concepts.concept_polygon.concept_definition"></a></span><a class="link" href="concept_polygon.html#geometry.reference.concepts.concept_polygon.concept_definition">Concept
          Definition</a>
        </h6>
<p>
          The Polygon Concept is defined as following:
        </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
              there must be a specialization of <code class="computeroutput"><span class="identifier">traits</span><span class="special">::</span><span class="identifier">tag</span></code>
              defining <code class="computeroutput"><span class="identifier">polygon_tag</span></code>
              as type
            </li>
<li class="listitem">
              there must be a specialization of <code class="computeroutput"><span class="identifier">traits</span><span class="special">::</span><span class="identifier">ring_type</span></code>
              defining the type of its exterior ring and interior rings as type
            </li>
<li class="listitem">
              this type defined by <code class="computeroutput"><span class="identifier">ring_type</span></code>
              must fulfill the <a class="link" href="concept_ring.html" title="Ring Concept">Ring
              Concept</a>
            </li>
<li class="listitem">
              there must be a specialization of <code class="computeroutput"><span class="identifier">traits</span><span class="special">::</span><span class="identifier">interior_type</span></code>
              defining the type of the collection of its interior rings as type;
              this collection itself must fulfill a Boost.Range Random Access Range
              Concept
            </li>
<li class="listitem">
              there must be a specialization of <code class="computeroutput"><span class="identifier">traits</span><span class="special">::</span><span class="identifier">exterior_ring</span></code>
              with two functions named <code class="computeroutput"><span class="identifier">get</span></code>,
              returning the exterior ring, one being const, the other being non const
            </li>
<li class="listitem">
              there must be a specialization of <code class="computeroutput"><span class="identifier">traits</span><span class="special">::</span><span class="identifier">interior_rings</span></code>
              with two functions named <code class="computeroutput"><span class="identifier">get</span></code>,
              returning the interior rings, one being const, the other being non
              const
            </li>
</ul></div>
<h6>
<a name="geometry.reference.concepts.concept_polygon.h2"></a>
          <span class="phrase"><a name="geometry.reference.concepts.concept_polygon.rules"></a></span><a class="link" href="concept_polygon.html#geometry.reference.concepts.concept_polygon.rules">Rules</a>
        </h6>
<p>
          Besides the Concepts, which are checks on compile-time, there are some
          other rules that valid polygons must fulfill. This follows the opengeospatial
          rules (see link above).
        </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
              Polygons are simple geometric objects (See also <a href="http://en.wikipedia.org/wiki/Simple_polygon" target="_top">wiki</a>
              but holes are allowed in Boost.Geometry polygons).
            </li>
<li class="listitem">
              If the polygons underlying <code class="computeroutput"><span class="identifier">ring_type</span></code>
              is defined as clockwise, the exterior ring must have the clockwise
              orientation, and any interior ring must be reversed w.r.t. the defined
              orientation (so: counter clockwise for clockwise exterior rings). If
              the <code class="computeroutput"><span class="identifier">ring_type</span></code> is defined
              counter clockwise, it is vice versa.
            </li>
<li class="listitem">
              If the polygons underlying <code class="computeroutput"><span class="identifier">ring_type</span></code>
              is defined as closed, all rings must be closed: the first point must
              be spatially equal to the last point.
            </li>
<li class="listitem">
              The interior is a connected point set.
            </li>
<li class="listitem">
              There should be no self intersections, but self tangencies (between
              exterior/interior rings) are allowed (as long as the interior is a
              connected point set.
            </li>
<li class="listitem">
              There should be no cut lines, spikes or punctures.
            </li>
<li class="listitem">
              The interior rings should be located within the exterior ring. Interior
              rings may not be located within each other.
            </li>
</ul></div>
<p>
          The algorithms such as intersection, area, centroid, union, etc. do not
          check validity. There will be an algorithm is_valid which checks for validity
          against these rules, at runtime, and which can be called (by the library
          user) before.
        </p>
<p>
          If the input is invalid, the output might be invalid too. For example:
          if a polygon which should be closed is not closed, the area will be incorrect.
        </p>
<h6>
<a name="geometry.reference.concepts.concept_polygon.h3"></a>
          <span class="phrase"><a name="geometry.reference.concepts.concept_polygon.available_models"></a></span><a class="link" href="concept_polygon.html#geometry.reference.concepts.concept_polygon.available_models">Available
          Models</a>
        </h6>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
              <a class="link" href="../models/model_polygon.html" title="model::polygon">polygon</a>
            </li>
<li class="listitem">
              a Boost.Polygon polygon_with_holes_data (requires <code class="computeroutput"><span class="preprocessor">#include</span>
              <span class="identifier">boost</span><span class="special">/</span><span class="identifier">geometry</span><span class="special">/</span><span class="identifier">geometries</span><span class="special">/</span><span class="identifier">adapted</span><span class="special">/</span><span class="identifier">boost_polygon</span><span class="special">/</span><span class="identifier">polygon</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></code>)
            </li>
</ul></div>
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      Bruno Lalande, Mateusz Loskot, Adam Wulkiewicz, Oracle and/or its affiliates<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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